Rational map (complex analysis)

A rational map of the Riemann sphere is a quotient of two complex polynomials, extended to infinity. A nonconstant map has a well-defined positive degree and is a branched covering of the sphere.

Rotations of the domain and target Riemann spheres act as Möbius transformations represented by SU(2) matrices. A combined symmetry requires the equivariance identity $R(gz)=hR(z)$ for the corresponding transformations. Symmetry of the Wronskian of a rational map alone is insufficient: it tests the ramification directions, not the entire map. Target rotations preserve the angular Jacobian of a rational map, so an equivariant map has a domain-invariant angular density.

For a reduced rational map $R=p/q$, its derivative is $W/q^2$ with $W=p^{\prime }q-p{q}^{\prime }$. Away from target-coordinate poles, the zeros of this Wronskian identify its ramification points of a holomorphic map; at a pole use the reciprocal coordinate $q/p$. A pole of order $m$ contributes ramification order $m-1$, and infinity must also be checked in local coordinates. For degree $N$ the Riemann-Hurwitz formula gives total ramification $2N-2$. The affine polynomial $W$ can have fewer finite zeros because some ramification lies at infinity. In the rational map approximation for Skyrmions, these directions have zero angular baryon density.

For a nonconstant rational map between round unit Riemann spheres, $R^{\ast }d\Omega =J_{R}d\Omega$. Counting inverse images with multiplicity gives $\int J_{R}d\Omega =4\pi \mathrm{deg}R$. The Jacobian determinant vanishes at ramification points of a holomorphic map, but the integral still records the global degree of a holomorphic map.